Pub Date : 2024-11-15DOI: 10.1016/j.fss.2024.109197
Michał Boczek, Tomasz Józefiak, Marek Kaluszka, Andrzej Okolewski
In this paper, we propose a new generalization of the classical discrete Choquet integral to the multivalued framework in terms of an admissible order that refines the natural partial order on the considered value set. The new Choquet-like operator takes as input a finite number of values of a given type, in particular real numbers, intervals, and vectors, and returns a single output value of the same type as the input values. We give necessary and sufficient conditions for the operator to be monotone with respect to the admissible order. We then provide a complete characterization of the Choquet-like operator as an aggregation function with respect to the admissible order and study its selected special cases.
{"title":"The Choquet-like operator with respect to an admissible order as a tool for aggregating multivalued data","authors":"Michał Boczek, Tomasz Józefiak, Marek Kaluszka, Andrzej Okolewski","doi":"10.1016/j.fss.2024.109197","DOIUrl":"10.1016/j.fss.2024.109197","url":null,"abstract":"<div><div>In this paper, we propose a new generalization of the classical discrete Choquet integral to the multivalued framework in terms of an admissible order that refines the natural partial order on the considered value set. The new Choquet-like operator takes as input a finite number of values of a given type, in particular real numbers, intervals, and vectors, and returns a single output value of the same type as the input values. We give necessary and sufficient conditions for the operator to be monotone with respect to the admissible order. We then provide a complete characterization of the Choquet-like operator as an aggregation function with respect to the admissible order and study its selected special cases.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109197"},"PeriodicalIF":3.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.fss.2024.109196
Xue-ping Wang, Yun-Mao Zhang
This article deals with the formula generated by a one-place function and a binary function . When the f is a strictly increasing function and F is a continuous, non-decreasing and associative function with neutral element in , the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the f is a non-decreasing surjective function and F is a non-decreasing associative function with neutral element in , we investigate the associativity of the formula.
本文讨论由一元函数 f:[0,1]→[0,1] 和二元函数 F:[0,1]2→[0,1] 生成的公式 f(-1)(F(f(x),f(y)))。当 f 是严格递增函数，F 是连续、非递减、关联函数且中性元素在 [0,1] 时，研究公式的以下代数和分析性质：幂等元素、连续性（左连续性/右连续性）、关联性和极限性质。研究了这些性质之间的关系。给出了公式成为三角形规范（即三角形 conorm）的一些必要条件和一些充分条件。特别是，给出了获得连续阿基米德三角形规范（或三角形 conorm）的必要条件和充分条件。当 f 是非递减的注入函数且 F 是非递减的关联函数且中性元素在 [0,1] 时，我们研究了公式的关联性。
{"title":"Some algebraic and analytical properties of a class of two-place functions","authors":"Xue-ping Wang, Yun-Mao Zhang","doi":"10.1016/j.fss.2024.109196","DOIUrl":"10.1016/j.fss.2024.109196","url":null,"abstract":"<div><div>This article deals with the formula <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>(</mo><mi>F</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo><mo>)</mo></math></span> generated by a one-place function <span><math><mi>f</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and a binary function <span><math><mi>F</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. When the <em>f</em> is a strictly increasing function and <em>F</em> is a continuous, non-decreasing and associative function with neutral element in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the <em>f</em> is a non-decreasing surjective function and <em>F</em> is a non-decreasing associative function with neutral element in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, we investigate the associativity of the formula.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109196"},"PeriodicalIF":3.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.fss.2024.109194
Linh Anh Nguyen
<div><div>Nondeterministic fuzzy transition systems (NFTSs) offer a robust framework for modeling and analyzing systems with inherent uncertainties and imprecision, which are prevalent in real-world scenarios. Wu et al. (2018) provided an algorithm for computing the crisp bisimilarity (the greatest crisp bisimulation) of a finite NFTS <span><math><mi>S</mi><mo>=</mo><mo>〈</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>δ</mi><mo>〉</mo></math></span>, with a time complexity of order <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> under the assumption that <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>. Qiao et al. (2023) provided an algorithm for computing the fuzzy bisimilarity (the greatest fuzzy bisimulation) of a finite NFTS <span><math><mi>S</mi></math></span> under the Gödel semantics, with a time complexity of order <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mi>l</mi><mo>)</mo></math></span> under the assumption that <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>, where <em>l</em> is the number of fuzzy values used in <span><math><mi>S</mi></math></span> plus 1. In this work, we provide efficient algorithms for computing the partition corresponding to the crisp bisimilarity of a finite NFTS <span><math><mi>S</mi></math></span>, as well as the compact fuzzy partition corresponding to the fuzzy bisimilarity of <span><math><mi>S</mi></math></span> under the Gödel semantics. Their time complexities are of the order <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mrow><mi>size</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo><mi>log</mi><mo></mo><mi>l</mi><mo>+</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>)</mo><mo>)</mo></math></span>, where <em>l</em> is the number of fuzzy values used in <span><math><mi>S</mi></math></span> plus 2. When <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>, this order is within <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>⋅</mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>⋅</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>)</mo></math></span>. The reduction of time complexity from <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><m
非确定性模糊过渡系统（NFTS）为建模和分析具有内在不确定性和不精确性的系统提供了一个稳健的框架，而这种不确定性和不精确性在现实世界的场景中非常普遍。Wu等人（2018）提供了一种计算有限NFTS S=〈S,A,δ〉的干脆二相似度（最大干脆二拟合）的算法，在|δ|≥|S||的假设下，时间复杂度为O(|S|4⋅|δ|2)阶。乔等人（2023）提供了一种在哥德尔语义下计算有限 NFTS S 的模糊二相似度（最大模糊二拟合）的算法，在假设|δ|≥|S|的情况下，时间复杂度为阶 O(|S|4⋅|δ|2⋅l)，其中 l 是 S 中使用的模糊值数加 1。在这项工作中，我们提供了计算与有限 NFTS S 的清晰相似性相对应的分区以及与哥德尔语义下 S 的模糊相似性相对应的紧凑模糊分区的高效算法。它们的时间复杂度为 O((size(δ)logl+|S|)log(|S|+|δ|))，其中 l 是 S 中使用的模糊值数加 2。当|δ≥||S|时，这个阶次在 O(|S|⋅|δ|⋅log2|δ|) 以内。时间复杂度从 O(|S|4⋅|δ|2) 和 O(|S|4⋅|δ|2⋅l) 降低到 O(|S|⋅|δ|⋅log2|δ|) 是这项工作的重大贡献。
{"title":"Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems","authors":"Linh Anh Nguyen","doi":"10.1016/j.fss.2024.109194","DOIUrl":"10.1016/j.fss.2024.109194","url":null,"abstract":"<div><div>Nondeterministic fuzzy transition systems (NFTSs) offer a robust framework for modeling and analyzing systems with inherent uncertainties and imprecision, which are prevalent in real-world scenarios. Wu et al. (2018) provided an algorithm for computing the crisp bisimilarity (the greatest crisp bisimulation) of a finite NFTS <span><math><mi>S</mi><mo>=</mo><mo>〈</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>δ</mi><mo>〉</mo></math></span>, with a time complexity of order <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> under the assumption that <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>. Qiao et al. (2023) provided an algorithm for computing the fuzzy bisimilarity (the greatest fuzzy bisimulation) of a finite NFTS <span><math><mi>S</mi></math></span> under the Gödel semantics, with a time complexity of order <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mi>l</mi><mo>)</mo></math></span> under the assumption that <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>, where <em>l</em> is the number of fuzzy values used in <span><math><mi>S</mi></math></span> plus 1. In this work, we provide efficient algorithms for computing the partition corresponding to the crisp bisimilarity of a finite NFTS <span><math><mi>S</mi></math></span>, as well as the compact fuzzy partition corresponding to the fuzzy bisimilarity of <span><math><mi>S</mi></math></span> under the Gödel semantics. Their time complexities are of the order <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mrow><mi>size</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo><mi>log</mi><mo></mo><mi>l</mi><mo>+</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>)</mo><mo>)</mo></math></span>, where <em>l</em> is the number of fuzzy values used in <span><math><mi>S</mi></math></span> plus 2. When <span><math><mo>|</mo><mi>δ</mi><mo>|</mo><mo>≥</mo><mo>|</mo><mi>S</mi><mo>|</mo></math></span>, this order is within <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>⋅</mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>⋅</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mo>|</mo><mi>δ</mi><mo>|</mo><mo>)</mo></math></span>. The reduction of time complexity from <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>⋅</mo><mo>|</mo><mi>δ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>S</mi><msup><mrow><m","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109194"},"PeriodicalIF":3.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.fss.2024.109195
Haiyu Pan , Jie Zhou , Yuming Lin , Yongzhi Cao
Fuzzy model checking, also called multi-valued model checking, has proved to be an effective technique in verifying properties of fuzzy systems. One important issue with fuzzy model checking, is that a model adopted in fuzzy model checking is frequently updated with small changes, and it is too costly to run a model-checking algorithm from scratch in response to every update. To address the issue, in this paper, we consider the incremental model-checking approach for fuzzy systems by making maximal use of previous model checking results or in other words, by minimizing unnecessary recomputation. The models of our study are fuzzy Kripke structures, which are a fuzzy counterpart of Kripke structures and used to describe fuzzy systems, while the properties of fuzzy systems are expressed using fuzzy computation tree logic, a fuzzy temporal logic derived from computation tree logic. The focus of the paper is on how to design incremental model-checking algorithms for two until-formulas which characterize the maximal or dually minimum constrained reachability properties with respect to fuzzy Kripke structures under transition insertions or deletions but not both. The feasibility of our approach is illustrated by an example arising from the path planning problem of mobile robots.
{"title":"Incremental model checking for fuzzy computation tree logic","authors":"Haiyu Pan , Jie Zhou , Yuming Lin , Yongzhi Cao","doi":"10.1016/j.fss.2024.109195","DOIUrl":"10.1016/j.fss.2024.109195","url":null,"abstract":"<div><div>Fuzzy model checking, also called multi-valued model checking, has proved to be an effective technique in verifying properties of fuzzy systems. One important issue with fuzzy model checking, is that a model adopted in fuzzy model checking is frequently updated with small changes, and it is too costly to run a model-checking algorithm from scratch in response to every update. To address the issue, in this paper, we consider the incremental model-checking approach for fuzzy systems by making maximal use of previous model checking results or in other words, by minimizing unnecessary recomputation. The models of our study are fuzzy Kripke structures, which are a fuzzy counterpart of Kripke structures and used to describe fuzzy systems, while the properties of fuzzy systems are expressed using fuzzy computation tree logic, a fuzzy temporal logic derived from computation tree logic. The focus of the paper is on how to design incremental model-checking algorithms for two until-formulas which characterize the maximal or dually minimum constrained reachability properties with respect to fuzzy Kripke structures under transition insertions or deletions but not both. The feasibility of our approach is illustrated by an example arising from the path planning problem of mobile robots.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109195"},"PeriodicalIF":3.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.fss.2024.109193
Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy
The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom and the infinitary rule to achieve strong completeness with respect to continuous t-norms.
{"title":"Strong completeness for the predicate logic of the continuous t-norms","authors":"Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy","doi":"10.1016/j.fss.2024.109193","DOIUrl":"10.1016/j.fss.2024.109193","url":null,"abstract":"<div><div>The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom <span><math><msup><mrow><mo>(</mo><mo>∀</mo><mi>x</mi><mi>ϕ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>↔</mo><mo>∀</mo><mi>x</mi><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the infinitary rule<span><span><span><math><mfrac><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><mi>α</mi><mi>&</mi><mi>β</mi><mo>)</mo></mrow></mfrac></math></span></span></span> to achieve strong completeness with respect to continuous t-norms.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109193"},"PeriodicalIF":3.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study emanates from a simple observation: as specified by Vapnik [37] in his study, an artificial neural network cannot generate a universal approximator if the aggregation function chosen to design the artificial neuron does not include non-linearity. The usual option is to follow a linear aggregation by a non-linear function, or so-called activation function. We wonder if this approach could be replaced by one using a natively non-linear aggregation function.
Among all of the available non-linear aggregation functions, here we are interested in aggregations based on weighted minimum and weighted maximum operations [8]. As these operators were originally developed within a possibility theory and fuzzy rule framework, such operators cannot be easily integrated into a neural network because the values that are usually considered belong to . For gradient descent based learning, a neuron must be an aggregation function derivable with respect to its inputs and synaptic weights, whose variables (synaptic weights, inputs and outputs) must all be signed real values. We thus propose an extension of weighted maximum based aggregation to enable this learning process. We show that such an aggregation can be seen as a combination of four Sugeno integrals. Finally, we compare this type of approach with the classical one.
{"title":"Combining thresholded real values for designing an artificial neuron in a neural network","authors":"Olivier Strauss , Agnès Rico , Jérôme Pasquet , Lionel Pibre","doi":"10.1016/j.fss.2024.109191","DOIUrl":"10.1016/j.fss.2024.109191","url":null,"abstract":"<div><div>This study emanates from a simple observation: as specified by Vapnik <span><span>[37]</span></span> in his study, an artificial neural network cannot generate a universal approximator if the aggregation function chosen to design the artificial neuron does not include non-linearity. The usual option is to follow a linear aggregation by a non-linear function, or so-called activation function. We wonder if this approach could be replaced by one using a natively non-linear aggregation function.</div><div>Among all of the available non-linear aggregation functions, here we are interested in aggregations based on weighted minimum and weighted maximum operations <span><span>[8]</span></span>. As these operators were originally developed within a possibility theory and fuzzy rule framework, such operators cannot be easily integrated into a neural network because the values that are usually considered belong to <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. For gradient descent based learning, a neuron must be an aggregation function derivable with respect to its inputs and synaptic weights, whose variables (synaptic weights, inputs and outputs) must all be signed real values. We thus propose an extension of weighted maximum based aggregation to enable this learning process. We show that such an aggregation can be seen as a combination of four Sugeno integrals. Finally, we compare this type of approach with the classical one.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109191"},"PeriodicalIF":3.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.fss.2024.109173
Marcin Kolenda , Filip Strobin , Kamil Wiśniewski
We consider versions of the chaos game algorithm for generating attractors of fuzzy iterated function systems (FIFSs for short). We show that a naive approach fails in a general setting of contractive FIFSs and we present its natural modifications that work in all cases. Our approach bases on certain modifications of orbits and shadings of a given FIFS, generated by disjunctive drivers. To complete the picture, we present also a version of the deterministic algorithm.
{"title":"Chaos game algorithm for fuzzy iterated function systems","authors":"Marcin Kolenda , Filip Strobin , Kamil Wiśniewski","doi":"10.1016/j.fss.2024.109173","DOIUrl":"10.1016/j.fss.2024.109173","url":null,"abstract":"<div><div>We consider versions of the chaos game algorithm for generating attractors of fuzzy iterated function systems (FIFSs for short). We show that a naive approach fails in a general setting of contractive FIFSs and we present its natural modifications that work in all cases. Our approach bases on certain modifications of orbits and shadings of a given FIFS, generated by disjunctive drivers. To complete the picture, we present also a version of the deterministic algorithm.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109173"},"PeriodicalIF":3.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.fss.2024.109176
Feiyue Shen , Haoyi Zhang , Wenhai Qi , Ju H. Park , Jun Cheng , Kaibo Shi
This paper studies the nonfragile anti-transitional-asynchrony fault tolerant control for discrete interval type-2 fuzzy semi-Markov jump systems with bounded dwell time. In contrast to the traditional Takagi-Sugeno (T-S) fuzzy model, this paper utilises the interval 2-type fuzzy model to represent the nonlinear discrete semi-Markov jump systems. The main novelty is to design an anti-transitional-asynchrony fault-tolerant control mechanism with controller gain fluctuations under the framework of IT2 fuzzy, fault-tolerant control, and transitional asynchrony, which reduces the conservatism to a certain extent. The transitional asynchrony is adopted, i.e., the controller switching lags behind the plant switching, and this lag is associated with the transition between the current mode and the next mode. By virtue of semi-Markov kernel method, the mean-square stability of the underlying system is achieved, overcoming the difficulties caused by the transitional asynchrony. In addition, taking into account that the actuator may encounter random faults during system operation, this paper applies a fault tolerant method in the design of nonfragile anti-transitional-asynchrony control to improve the fault tolerance of the system. Finally, a tunnel circuit model verifies the effectiveness of the designed control method.
{"title":"Nonfragile anti-transitional-asynchrony fault tolerant control for IT2 fuzzy semi-Markov jump systems with actuator failures","authors":"Feiyue Shen , Haoyi Zhang , Wenhai Qi , Ju H. Park , Jun Cheng , Kaibo Shi","doi":"10.1016/j.fss.2024.109176","DOIUrl":"10.1016/j.fss.2024.109176","url":null,"abstract":"<div><div>This paper studies the nonfragile anti-transitional-asynchrony fault tolerant control for discrete interval type-2 fuzzy semi-Markov jump systems with bounded dwell time. In contrast to the traditional Takagi-Sugeno (T-S) fuzzy model, this paper utilises the interval 2-type fuzzy model to represent the nonlinear discrete semi-Markov jump systems. The main novelty is to design an anti-transitional-asynchrony fault-tolerant control mechanism with controller gain fluctuations under the framework of IT2 fuzzy, fault-tolerant control, and transitional asynchrony, which reduces the conservatism to a certain extent. The transitional asynchrony is adopted, i.e., the controller switching lags behind the plant switching, and this lag is associated with the transition between the current mode and the next mode. By virtue of semi-Markov kernel method, the mean-square stability of the underlying system is achieved, overcoming the difficulties caused by the transitional asynchrony. In addition, taking into account that the actuator may encounter random faults during system operation, this paper applies a fault tolerant method in the design of nonfragile anti-transitional-asynchrony control to improve the fault tolerance of the system. Finally, a tunnel circuit model verifies the effectiveness of the designed control method.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109176"},"PeriodicalIF":3.2,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.fss.2024.109171
Siyu Guo, Yingnan Pan, Zhechen Zhu
This paper considers the prescribed performance fuzzy adaptive tracking control problem of multiagent systems under a chaos-based privacy-preserving mechanism and a learning-enabled event-triggered mechanism. Initially, a chaos-based mask function is constructed, which is related to the chaotic states in the Lorentz system. The utilization of chaos adds unpredictability and full randomness to the mask function, which greatly reduces the risk of privacy leakage. Additionally, two value functions are designed as inputs of the fully connected neural network, and the fully connected neural network is used to predict the parameter value in the event-triggered mechanism, which effectively enhances the flexibility of the proposed learning-enabled event-triggered mechanism. Furthermore, in the process of controller design, by employing an error transformed function, the system errors are stabilized within the prescribed performance boundaries. Finally, a simulation example is provided to validate the effectiveness of the proposed control scheme.
{"title":"Learning-enabled event-triggered fuzzy adaptive control of multiagent systems with prescribed performance: A chaos-based privacy-preserving method","authors":"Siyu Guo, Yingnan Pan, Zhechen Zhu","doi":"10.1016/j.fss.2024.109171","DOIUrl":"10.1016/j.fss.2024.109171","url":null,"abstract":"<div><div>This paper considers the prescribed performance fuzzy adaptive tracking control problem of multiagent systems under a chaos-based privacy-preserving mechanism and a learning-enabled event-triggered mechanism. Initially, a chaos-based mask function is constructed, which is related to the chaotic states in the Lorentz system. The utilization of chaos adds unpredictability and full randomness to the mask function, which greatly reduces the risk of privacy leakage. Additionally, two value functions are designed as inputs of the fully connected neural network, and the fully connected neural network is used to predict the parameter value in the event-triggered mechanism, which effectively enhances the flexibility of the proposed learning-enabled event-triggered mechanism. Furthermore, in the process of controller design, by employing an error transformed function, the system errors are stabilized within the prescribed performance boundaries. Finally, a simulation example is provided to validate the effectiveness of the proposed control scheme.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109171"},"PeriodicalIF":3.2,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.fss.2024.109189
Samira Fatemi, Ildar Sadeqi, Fridoun Moradlou
We give the concept of subdifferentiability for a fuzzy n-cell number-valued function. Then we state some theorems related to the subdifferentiability based on the new definition. Finally, we present some applications focused on optimization problems, including the Lagrangian dual problem and the composite minimization problem.
我们给出了模糊 n 单元数值函数的次微分性概念。然后，我们根据新定义阐述了与次微分性相关的一些定理。最后，我们介绍了优化问题的一些应用，包括拉格朗日对偶问题和复合最小化问题。
{"title":"Subdifferential of fuzzy n-cell number-valued functions and its applications in optimization problems","authors":"Samira Fatemi, Ildar Sadeqi, Fridoun Moradlou","doi":"10.1016/j.fss.2024.109189","DOIUrl":"10.1016/j.fss.2024.109189","url":null,"abstract":"<div><div>We give the concept of subdifferentiability for a fuzzy <em>n</em>-cell number-valued function. Then we state some theorems related to the subdifferentiability based on the new definition. Finally, we present some applications focused on optimization problems, including the Lagrangian dual problem and the composite minimization problem.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109189"},"PeriodicalIF":3.2,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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